When can I dodge damage with Temporal Breach?

It depends on where the damage is coming from — breaching dodges damage from a spell or triggered ability that's on the chain, but it can't dodge combat damage, and it can't dodge damage that's already been marked.

The working dodge: play Temporal Breach in reaction to the damage spell/ability. Breach resolves first, the unit is banished and replayed as a brand-new object, so the pending spell's target has changed zones to a non-board zone and back — it's no longer a legal target and the damage instruction is ignored (Rule 359.3.e.4). Same trick saves a unit from a pending kill (a Temporary trigger, a delayed "kill it" effect), because that ability can't read the old object. It also wipes any damage already marked and any "this turn" buffs/statuses, since the new object starts clean.

Combat damage is not dodgeable. When the Showdown closes it rolls straight into assigning and dealing damage with no priority window (467.2, 467.3 skips FEPR), so there's nothing to react to. Breaching your own unit mid-Showdown does clear its marked damage, but it comes back at the same battlefield and is still there when damage is assigned — that's a heal, not a dodge. And once lethal damage has landed, the unit dies in the very next cleanup before anyone gets priority to breach at all (FAQ #12271).

Two practical gates: the hidden play is free only if you hid the Breach on a previous turn and the target unit is at the battlefield where it's hidden; otherwise you're casting a 2-energy spell at Action speed in an Open State, which won't be available mid-chain.